Restricted product
In mathematics, the restricted product is a construction in the theory of topological groups.
Let be an index set; a finite subset of. If is a locally compact group for each, and is an open compact subgroup for each, then the restricted product
is the subset of the product of the 's consisting of all elements such that for all but finitely many.
This group is given the topology whose basis of open sets are those of the form
where is open in and for all but finitely many.
One can easily prove that the restricted product is itself a locally compact group. The best known example of this construction is that of the adele ring and idele group of a global field.
